South Korea proposed restarting a much-missed inter-Korean project, in a fresh attempt at rapprochement.

South Korean President Park Geun-hye said she will seek to have a reunion of war-divided families held by the Lunar New Year holiday at the end of the month—a traditional time for family gatherings. Seoul’s Unification Ministry, which handles inter-Korean affairs, said it sent Pyongyang a request Monday afternoon that the two sides hold a preliminary meeting on the subject Friday. North Korea hasn’t yet responded.

First held in 1985, the emotionally charged mass reunions of families divided by the 1950-53 Korean War took place regularly on both sides of the border and by video chat between 2000 and 2010, when the North’s deadly artillery attack on a South Korean island led to an interruption. Though tens of thousands of people have attended reunions, the waiting list just on the South Korean side numbers 70,000, and experts estimate that about 2,000 people die every year without realizing their dream.

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The most recent family meeting had been planned for late September, but Pyongyang pulled the plug a few days before, blaming the indefinite postponement on Seoul’s “confrontation racket.”

That came despite the positive sign of resumed operations at the jointly run Kaesong industrial park, just north of the border. Last April, North Korea withdrew all its workers from the factories during a flare-up in tensions, forcing a shutdown until September’s restart.

But then Mr. Kim in his annual new-year speech last week expressed a wish for warmer ties with Seoul. The South questioned the authenticity of those sentiments, drawing a spate of hostile rhetoric in the North’s state media.

Ms. Park offered the reunion proposal during her first press conference since her inauguration early last year. She also stressed reunification of the two Koreas as one of her priorities for the remainder of her term, which ends in early 2018.

“I’m aware some South Koreans saying reunification will be too costly and questioning whether it’s necessary,” said Ms. Park. But she argued that it would provide a boost to the combined economy.